History of impact of non-Euclidean geometry on math, philosophy, and the public

Question

I'm interested in the impact of the discovery of non-Euclidean geometry on math, philosophy, and the attitudes of the general public.

I don't know anything about how things changed right after the discoveries in the 1840's. My knowledge starts at Cantor and continues forward.

By way of examples: What did mathematicians think about the loss of certainty in their field?

How did philosophers process this development?

As regards the public; my understanding is that this when postmodernism and the culture wars began. If there are mutually inconsistent geometries, then perhaps we're each entitled to our own subjective reality. How did the world get from there to here? Is that a generally accepted view of the social impact? Where can I read more about it?

I've heard of Kline's book on the Loss of Certainty. Is that something I should read?

Thanks much.

Answer 1

Regarding :

• Morris Kline, Mathematics: The Loss of Certainty (1982)

the answer is : yes; Ch.IV The First Debacle : The Withering of Truth is dedicated to non-Eucliidean geometry, and it is worth to be read.

More details into :

• BA Rosenfeld, A history of non-euclidean geometry (1988)

• Jeremy Gray, Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century (2007)

• Jeremy Gray, Plato's Ghost: The Modernist Transformation of Mathematics (2008).

For a more "philosophical" point of view, see :

• Roberto Torretti, Philosophy of Geometry from Riemann to Poincaré (1978).

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For an overview, see also Nineteenth Century Geometry, Kant's Views on Space and Time, Hermann von Helmholtz and Henri Poincaré.

Answer 2

Have you taken a look at Richard Trudeau's The non-Euclidean revolution (with an introduction by H. S. M. Coxeter. Birkhäuser Boston, Inc., Boston, MA, 1987. xiv+269 pp. ISBN: 0-8176-3311-1)?

The first two paragraphs of the review of this book, which K. Strubecker contributed to MathSciNet, read thus:

Starting from a very detailed, critical overview of plane geometry as axiomatically based by Euclid in his Elements, the author, in this remarkable book, describes in an incomparable way the fascinating path taken by the geometry of the plane in its historical evolution from antiquity up to the discovery of non-Euclidean geometry. This discovery, characterized by the names of Gauss, Bolyai and Lobachevskiĭ, signified a revolution not just for geometry; the philosophical views of space—as shaped predominantly by Kant and generally accepted—and also epistemology underwent an unforeseen upheaval. This "non-Euclidean revolution'', in all its aspects, is described very strikingly here.

The book begins with a detailed critical discussion of Euclidean explanations of concepts, definitions and axioms. The fifth axiom, in Euclid's version not very transparent, then, in the intuitive formulation named after John Playfair, appeared very much clearer and convincing. The many futile attempts at proving Euclid's "parallel axiom'' and the final solution of the parallel problem through the discovery of non-Euclidean geometry are described in detail. Finally, the relative freedom from contradiction in non-Euclidean geometry is proved. Much attention is paid to thus newly acquired horizons of knowledge and their influence on problems in natural philosophy and particularly in physics...

Seems to me that this book might be right up your alley!